Solve Square Root of Fraction: Simplifying √(81/9)

Q: Can I take the square root of the top and bottom separately?

Yes, you can use the property $ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} $, but it's usually easier to simplify the fraction first. For $ \sqrt{\frac{81}{9}} $, simplifying to $ \sqrt{9} $ is much simpler!

Q: How do I know if 9 is a perfect square?

A perfect square is a number that equals some integer times itself. Since $ 3 \times 3 = 9 $, we know that $ \sqrt{9} = 3 $. Common perfect squares include 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

Q: Why is the answer 3 instead of 9?

Remember that $ \sqrt{9} = 3 $, not 9! The square root asks 'what number times itself gives 9?' That number is 3, because $ 3^2 = 3 \times 3 = 9 $.

Q: Do I always need to simplify fractions before taking square roots?

It's not required, but it's highly recommended! Simplifying first often reveals perfect squares that make the problem much easier to solve and reduces calculation errors.

Square Root Properties with Fraction Simplification

Complete the following exercise:

$\sqrt{\frac{81}{9}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve the following problem

00:03 Calculate 81 divided by 9

00:07 Break down 9 into 3 squared

00:11 The root of any number (A) squared cancels out the square

00:14 Apply this formula to our exercise, and proceed to cancel out the square

00:17 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the following exercise:

$\sqrt{\frac{81}{9}}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Simplify the fraction inside the square root.
Step 2: Apply the square root quotient property.
Step 3: Perform the necessary calculations and simplify if needed.

Let's work through each step:

Step 1: Simplify the fraction $\frac{81}{9}$ . This simplifies to $9$ , as dividing $81$ by $9$ gives us $9$ .

Step 2: Apply the square root property. We need to calculate $\sqrt{9}$ .

Step 3: Calculate the square root. $\sqrt{9} = 3$ , since $3 \times 3 = 9$ .

Therefore, the solution to the problem is $3$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Simplify the fraction inside the square root first
Technique: $\frac{81}{9} = 9$ , then $\sqrt{9} = 3$
Check: Verify $3^2 = 9$ and $9 = \frac{81}{9}$ ✓

Common Mistakes

Avoid these frequent errors

Taking square root of numerator and denominator separately
Don't calculate $\frac{\sqrt{81}}{\sqrt{9}} = \frac{9}{3} = 3$ as separate steps! While this gives the correct answer here, it's unnecessarily complicated and can lead to errors with more complex fractions. Always simplify the fraction first, then take the square root.

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt{\frac{2}{4}}=$

FAQ

Everything you need to know about this question

Can I take the square root of the top and bottom separately?

Yes, you can use the property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ , but it's usually easier to simplify the fraction first. For $\sqrt{\frac{81}{9}}$ , simplifying to $\sqrt{9}$ is much simpler!

What if the fraction doesn't simplify to a perfect square?

If the simplified fraction isn't a perfect square, you can either leave it as a square root or use the property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ to separate the numerator and denominator.

How do I know if 9 is a perfect square?

A perfect square is a number that equals some integer times itself. Since $3 \times 3 = 9$ , we know that $\sqrt{9} = 3$ . Common perfect squares include 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

Why is the answer 3 instead of 9?

Remember that $\sqrt{9} = 3$ , not 9! The square root asks 'what number times itself gives 9?' That number is 3, because $3^2 = 3 \times 3 = 9$ .

Do I always need to simplify fractions before taking square roots?

It's not required, but it's highly recommended! Simplifying first often reveals perfect squares that make the problem much easier to solve and reduces calculation errors.

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